I am interested in the standard name for the following weak form of $k$-differentiability.

Definition. A function $f:\mathbb R\to\mathbb R$ is called Taylor $k$-differentiable at a point $x_0$ if there are real numbers $a_0,\dots,a_k$ such that $$f(x)=\sum_{n=0}^k\tfrac1{n!}a_n(x-x_0)^n+o(|x-x_0|^n).$$The numbers $a_0,\dots,a_k$ are unique (if exist) and can be called the derivatives $f^{(0)}(x_0),\dots,f^{(k)}(x_0)$ of $f$ at $x_0$.

The Taylor formula says that each $k$-differentiable function at a point $x_0$ is Taylor $k$-differentiable. The converse is not true: the Dirichlet-like function $$f(x)=\begin{cases}x^{k+1}&\mbox{if $x$ is rational};\\ 0&\mbox{if $x$ is irraional} \end{cases} $$ is Taylor $k$-differentiable at zero (with all derivatives $f^{(0)}(0)=\dots=f^{(k)}(0)=0$) but is discontinuous at all non-zero points, so is not $k$-differentiable in the standard sense.

So, my question:

Has the Taylor $k$-differentiability some standard name, accepted in the literature? If yes, what is a suitable reference? Thanks.

I am interested in the standard name for the following weak form of $k$-differentiability.

Definition. A function $f:\mathbb R\to\mathbb R$ is called Taylor $k$-differentiable at a point $x_0$ if there are real numbers $a_0,\dots,a_k$ such that $$f(x)=\sum_{n=0}^k\tfrac1{n!}a_n(x-x_0)^n+o(|x-x_0|^k).$$The numbers $a_0,\dots,a_k$ are unique (if exist) and can be called the derivatives $f^{(0)}(x_0),\dots,f^{(k)}(x_0)$ of $f$ at $x_0$.

The Taylor formula says that each $k$-differentiable function at a point $x_0$ is Taylor $k$-differentiable. The converse is not true: the Dirichlet-like function $$f(x)=\begin{cases}x^{k+1}&\mbox{if $x$ is rational};\\ 0&\mbox{if $x$ is irraional} \end{cases} $$ is Taylor $k$-differentiable at zero (with all derivatives $f^{(0)}(0)=\dots=f^{(k)}(0)=0$) but is discontinuous at all non-zero points, so is not $k$-differentiable in the standard sense.

So, my question:

Has the Taylor $k$-differentiability some standard name, accepted in the literature? If yes, what is a suitable reference? Thanks.